Effective resistance is a distance between vertices of a graph that is both theoretically interesting and useful in applications. We study a variant of effective resistance called the biharmonic distance. While the effective resistance measures how well-connected two vertices are, we prove several theoretical results supporting the idea that the biharmonic distance measures how important an edge is to the global topology of the graph. Our theoretical results connect the biharmonic distance to well-known measures of connectivity of a graph like its total resistance and sparsity. Based on these results, we introduce two clustering algorithms using the biharmonic distance. Finally, we introduce a further generalization of the biharmonic distance that we call the $k$-harmonic distance. We empirically study the utility of biharmonic and $k$-harmonic distance for edge centrality and graph clustering.